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*P60791A0128*P60791A©2019 Pearson Education Ltd.1/1/1/1/1/C2/

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Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions• Use black ink or ball-point pen.• If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).• Fill in the boxes at the top of this page with your name, centre number and candidate number.• Answer all questions and ensure that your answers to parts of questions are clearly labelled.• Answer the questions in the spaces provided – there may be more space than you need.• You should show sufficient working to make your methods clear. Answers without working may not gain full credit.• Inexact answers should be given to three significant figures unless otherwise stated.

Information• A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.• There are 12 questions in this question paper. The total mark for this paper is 75.• The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice• Read each question carefully before you start to answer it.• Try to answer every question.• Check your answers if you have time at the end.• If you change your mind about an answer, cross it out and put your new answer and any working underneath.

You must have:Mathematical Formulae and Statistical Tables (Lilac), calculator

MathematicsAdvanced SubsidiaryPure Mathematics P1

Paper Reference WMA11/01Morning (Time: 1 hour 30 minutes)

Tuesday 8 January 2019

Pearson Edexcel InternationalAdvanced Level

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Answer ALL questions. Write your answers in the spaces provided.

1. Find

∫ − + 23 12 53 3x x xd

simplifying your answer.(4)

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Question 1 continued

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___________________________________________________________________________ Q1

(Total 4 marks)

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2. Given33

27 34x

y =

find y as a simplified function of x.(3)

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Question 2 continued

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(Total 3 marks)

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3. The line l1 has equation 3x + 5y − 7 = 0

(a) Find the gradient of l1(2)

The line l2 is perpendicular to l1 and passes through the point (6, −2).

(b) Find the equation of l2 in the form y = mx + c , where m and c are constants.(3)

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(Total 5 marks)

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4.y

x

R

C

l1

O

l2

Figure 1

Figure 1 shows a line l1 with equation 2y = x and a curve C with equation y = 2x − 18

x2

The region R, shown unshaded in Figure 1, is bounded by the line l1 , the curve C and a line l2

Given that l2 is parallel to the y-axis and passes through the intercept of C with the positive x-axis, identify the inequalities that define R.

(3)

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Question 4 continued

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(Total 3 marks)

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5.y

xO

P

Figure 2

Figure 2 shows a plot of part of the curve with equation y = cos 2x with x being measured in radians.

The point P, shown on Figure 2, is a minimum point on the curve.

(a) State the coordinates of P.(2)

A copy of Figure 2, called Diagram 1, is shown at the top of the next page.

(b) Sketch, on Diagram 1, the curve with equation y = sin x(2)

(c) Hence, or otherwise, deduce the number of solutions of the equation

(i) cos 2x = sin x  that lie in the region  0  x  20π

(ii) cos 2x = sin x  that lie in the region  0  x 21π(2)

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Question 5 continued

y

xO

P

Diagram 1

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___________________________________________________________________________ Q5

(Total 6 marks)

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6. (Solutions based entirely on graphical or numerical methods are not acceptable.)

Given

f (x) = 2x52 − 40x + 8 x > 0

  (a)  solve the equation  f ʹ (x) = 0(4)

  (b)  solve the equation  f ʺ (x) = 5(3)

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Question 6 continued

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(Total 7 marks)

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7.

A B

C

D

35°

Not to scale

Figure 3

Figure 3 shows the design for a structure used to support a roof.

The structure consists of four wooden beams, AB, BD, BC and AD.

Given AB = 6.5 m,  BC = BD = 4.7 m and angle BAC = 35°

(a) find, to one decimal place, the size of angle ACB,(3)

(b) find, to the nearest metre, the total length of wood required to make this structure.(3)

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Question 7 continued

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___________________________________________________________________________ Q7

(Total 6 marks)

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8.

y

O

y = 4

y = f ( x )

(4, 9)

(0, 6)

(−3, 0) x

Figure 4

The curve C with equation y = f ( x )  is shown in Figure 4.

The curve C

• has a single turning point, a maximum at (4, 9)

•  crosses the coordinate axes at only two places, (−3, 0) and (0, 6)

• has a single asymptote with equation y = 4

as shown in Figure 4.

(a) State the equation of the asymptote to the curve with equation y = f (− x ).(1)

(b) State the coordinates of the turning point on the curve with equation y = f 14

x

.

(1)

Given that the line with equation y = k, where k is a constant, intersects C at exactly one point,

(c) state the possible values for k.(2)

The curve C is transformed to a new curve that passes through the origin.

(d) (i) Given that the new curve has equation y = f ( x ) − a , state the value of the constant a.

(ii) Write down an equation for another single transformation of C that also passes through the origin.

(2)

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___________________________________________________________________________ Q8

(Total 6 marks)

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9. The equation

3x + 5 = −2x + c

where c is a constant, has no real roots.

Find the range of possible values of c.(7)

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Question 9 continued

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___________________________________________________________________________ Q9

(Total 7 marks)

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10. A sector AOB, of a circle centre O, has radius r cm and angle θ radians.

Given that the area of the sector is 6 cm2 and that the perimeter of the sector is 10 cm, 

(a) show that

3θ 2 − 13θ + 12 = 0(4)

(b) Hence find possible values of r and θ.(3)

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Question 10 continued

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___________________________________________________________________________ Q10

(Total 7 marks)

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11. (a) On Diagram 1 sketch the graphs of

(i) y = x(3 − x)

(ii) y = x(x − 2)(5 − x)

showing clearly the coordinates of the points where the curves cross the coordinate axes.

(4)

(b) Show that the x coordinates of the points of intersection of

y = x(3 − x) and y = x(x − 2)(5 − x)

are given by the solutions to the equation x(x2 − 8x + 13) = 0(3)

The point P lies on both curves. Given that P lies in the first quadrant,

(c) find, using algebra and showing your working, the exact coordinates of P.(5)

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Question 11 continued

y

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___________________________________________________________________________ Q11

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12. The curve with equation y = f ( x ), x > 0, passes through the point P(4, −2).

Given that

dd

yx

x x x= − −3 1012

(a) find the equation of the tangent to the curve at P, writing your answer in the form y = mx + c , where m and c are integers to be found.

(4)

  (b) Find  f ( x ).(5)

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Question 12 continued

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Question 12 continued

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TOTAL FOR PAPER IS 75 MARKS

Q12

(Total 9 marks)